reserve x,x1,x2,x3 for Real;

theorem
  cos(x)<>0 & sin(x)<>0 implies cosec(2*x)=sec(x)*cosec(x)/2 & cosec(2*x
  )=(tan(x)+cot(x))/2
proof
  assume that
A1: cos(x)<>0 and
A2: sin(x)<>0;
A3: cosec(2*x)=1/((2*tan(x))/(1+(tan(x))^2)) by A1,Th6
    .=(1+(tan(x))^2)/(2*tan(x)) by XCMPLX_1:57
    .=(1+(tan(x))^2)/tan(x)/2 by XCMPLX_1:78
    .=(1/(sin(x)/cos(x))+(tan(x))^2/tan(x))/2 by XCMPLX_1:62
    .=(cot(x)+tan(x)*tan(x)/tan(x))/2 by XCMPLX_1:57
    .=(cot(x)+tan(x))/2 by A1,A2,XCMPLX_1:50,89;
  cosec(2*x)=1/((2*tan(x))/(1+(tan(x))^2)) by A1,Th6
    .=(1+(tan(x))^2)/(2*tan(x)) by XCMPLX_1:57
    .=(sec(x))^2/(2*tan(x)) by A1,Th11
    .=sec(x)*sec(x)/tan(x)/2 by XCMPLX_1:78
    .=sec(x)*(sec(x)/(sin(x)/cos(x)))/2 by XCMPLX_1:74
    .=sec(x)*(sec(x)*cos(x)/sin(x))/2 by XCMPLX_1:77
    .=sec(x)*cosec(x)/2 by A1,XCMPLX_1:106;
  hence thesis by A3;
end;
